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A319800
Numbers k such that Sum_{d|k} nphi(d) = k where the sum is over nonunitary divisors of k and nphi(k) is the nonunitary totient function (A254503).
0
3960, 5220, 1873080, 6733440, 8447040, 18685336320, 255306083760, 341863562880, 357274165248, 765899971200, 1018887932160
OFFSET
1,1
COMMENTS
Ligh and Wall found the first 5 terms and also the terms 18685336320, 341863562880, 357174165248, 1018887932160, 20993596382889043200. They showed that each term has a powerful part with at least 2 distinct prime factors, and conjectured that it is only even.
REFERENCES
Jozsef Sandor and Borislav Crstici, Handbook of Number Theory II, Kluwer Academic Publishers, 2004, Chapter 3, p. 287.
LINKS
Steve Ligh and Charles R. Wall, Functions of Nonunitary Divisors, Fibonacci Quarterly, Vol. 25 (1987), pp. 333-338.
MATHEMATICA
rad[n_] := Times @@ First /@ FactorInteger[n]; powerFree[n_] := Denominator[ n/rad[n]^2 ]; powerPart[n_] := n/powerFree[n]; nuphi[n_] := powerFree[ n ] * EulerPhi[powerPart[n]]; ndiv[n_] := Block[{d = Divisors[n]}, Select[d, GCD[#, n/#] > 1 &]]; a[n_] := Module[{d = ndiv[n]}, Total@Map[nuphi, d]]; s={}; Do[ If[a[n] == n, AppendTo[s, n]], {n, 1, 10^8}]; s
PROG
(PARI) nphi(n) = sumdiv(n, d, if(gcd(n/d, d) == 1, moebius(d)^2 * eulerphi(n/d)));
isok(n) = sumdiv(n, d, if(gcd(n/d, d) != 1, nphi(d))) == n; \\ Michel Marcus, Sep 28 2018
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Amiram Eldar, Sep 28 2018
EXTENSIONS
a(6)-a(9) from Giovanni Resta, Sep 29 2018
a(10)-a(11) from Giovanni Resta, Oct 11 2018
STATUS
approved